Tangents

[DanDan]

OK so my question is 'Show the lengths of the tangents from the point, P, (5,7) to the circle x²+y²-2x-4y-4=0

And this is how far I've got

[x²-2x+(-1)²]+[y²-4y+(-2)²=4+(-1)²+(-2)²
(x-1)²+(y-2)²=9

Center = (1,2) and Radius = 3

A is the point of contact of one of the tangents, so PA is the length of the tangent and CA is the radius.

CP²=(1+5)²+(2+7)²​=117

CAP=90 degrees so PA²=CP²-CA²
PA²=117-9=108

Is all of that right so far? because I feel like some of it is wrong when I start to use Pythagoras.

 

[pappus]

OK so my question is 'Show the lengths of the tangents from the point, P, (5,7) to the circle x²+y²-2x-4y-4=0

And this is how far I've got

[x²-2x+(-1)²]+[y²-4y+(-2)²=4+(-1)²+(-2)²
(x-1)²+(y-2)²=9

Center = (1,2) and Radius = 3

A is the point of contact of one of the tangents, so PA is the length of the tangent and CA is the radius.

CP²=(1+5)²+(2+7)²​=117

CAP=90 degrees so PA²=CP²-CA²
PA²=117-9=108

Is all of that right so far? \(\displaystyle <--- no\) because I feel like some of it is wrong when I start to use Pythagoras.

1. Midpoint, length of radius are OK.

2. Draw a sketch!

3. The distance between the center and point P is: \(\displaystyle |\overline{CP}|=\sqrt{(5-1)^2+(7-2)^2}\)

4. The length of PA is determined by Pythagorean theorem:

\(\displaystyle (\sqrt{41})^2 - 3^2 = (|\overline{PA}|)^2\)

5. For confirmation only: I've got \(\displaystyle |\overline{PA}| = 4 \sqrt{2}\)